This invention relates to the field of fluorescence spectroscopy, and more particularly to a method for determining characteristic physical quantities of fluorescent molecules or other particles present in a sample.
The primary data of an experiment in fluorescence correlation spectroscopy (FCS) is a sequence of photon counts detected from a microscopic measurement volume. An essential attribute of the fluorescence correlation analysis is the calculation of the second order autocorrelation function of photon detection. This is a way how a stochastic function (of photon counts) is transformed into a statistical function having an expected shape, serving as a means to estimate some parameters of the sample. However, the calculation of the autocorrelation function is not the only way for extracting information about the sample from the sequence of photon counts. Further approaches are based on moment analysis and analysis of the distribution of the number of photon counts per given time interval (Qian and Elson, Proc. Natl. Acad. Sci. USA, 87:5479-483, 1990; Qian and Elson, Biophys. J. 57:375-380, 1990).
The intensity of fluorescence detected from a particle within a sample is not uniform but depends on the coordinates of the particle with respect to the focus of the optical system. Therefore, a reliable interpretation of measurements should account for the geometry of the illuminated measurement volume. Even though the calculation of a theoretical distribution of the number of photo counts is more complex for a bell-shaped profile than for a rectangular one, the distribution of the number of photon counts sensitively depends on values of the concentration and the specific brightness of fluorescent species, and therefore, the measured distributions of the number of photon counts can be used for sample analysis. The term xe2x80x9cspecific brightnessxe2x80x9d denotes the mean count rate of the detector from light emitted by a particle of given species situated in a certain point in the sample, conventionally in the point where the value of the spatial brightness profile function is unity.
The first realization of this kind of analysis was demonstrated on the basis of moments of the photon count number distribution (Qian and Elson, Proc. Natl. Acad. Sci. USA, 87: 5479-483, 1990). The k-th factorial moment of the photon count number distribution P(n) is defined as                               F          k                =                              ∑            n                    ⁢                                                    n                !                                                              (                                      n                    -                    k                                    )                                !                                      ⁢                                          P                ⁡                                  (                  n                  )                                            .                                                          (        1        )            
In turn, factorial moments are closely related to factorial cumulants,                                           F            k                    =                                    ∑                              l                =                0                                            k                -                l                                      ⁢                                          C                l                                  k                  -                  1                                            ⁢                              K                                  k                  -                  l                                            ⁢                              F                i                                                    ,                  
                ⁢        or                            (        2        )                                          K          l                =                              F            k                    -                                    ∑                              i                =                1                                            k                -                1                                      ⁢                                          C                l                                  k                  -                  1                                            ⁢                              K                                  k                  -                  l                                            ⁢                                                F                  i                                .                                                                        (        3        )            
(C1ks are binomial coefficients, and Kks are cumulants). The basic expression used in moment analysis, derived for ideal solutions, relate k-th order cumulant to concentrations (c1) and specific brightness values (q1).                               K          k                =                              χ            k                    ⁢                                    ∑              i                        ⁢                                                                                c                    i                                    ⁡                                      (                                                                  q                        i                                            ⁢                      T                                        )                                                  k                            .                                                          (        4        )            
Here, Xk is the k-th moment of the relative spatial brightness profile B(r):                               χ          k                =                              ∫                          (              V              )                                ⁢                                                    B                k                            ⁡                              (                r                )                                      ⁢                                          ⅆ                V                            .                                                          (        5        )            
Usually in FCS, the unit of volume and the unit of B are selected which yield X1=X2=1. After selecting this convention, concentrations in the equations are dimensionless, expressing the mean number of particles per measurement volume, and the specific brightness of any species equals the mean count rate from a particle if situated in the focus divided by the numeric value of B(0). The value of this constant is characteristic of optical equipment. It can be calculated from estimated parameters of the spatial intensity profile (see below). Qian and Elson used experimental values of the first three cumulants to determine unknown parameters of the sample. The number of cumulants which can be reliably determined from experiments is usually three to four. This sets a limit to the applicability of the moment analysis.
The idea behind the so-called fluorescence intensity distribution analysis (FIDA), which has in detail been described in the international patent application PCT/EP 97/05619 (international publication number WO 98/16814), can be well understood by imagining an ideal case when a measurement volume is uniformly illuminated and when there is almost never more than a single particle illuminated at a time, similar to the ideal situation in cell sorters. Under these circumstances, each time when a particle enters the measurement volume, fluorescence intensity jumps to a value corresponding to the brightness of a given type of particles. Naturally, the probability that this intensity occurs at an arbitrary time moment equals the product of the concentration of a given species and the size of the measurement volume. Another fluorescent species which may be present in the sample solution produces intensity jumps to another value characteristic of this other species. In summary, the distribution of light intensity is in a straightforward way determined by the values of concentration and specific brightness of each fluorescent species in the sample solution.
It is assumed that the light intensity reaching the detector from a particle as a function of coordinates of the particle is constant over the whole measurement volume, and zero outside it. Also, it is assumed that the diffusion of a fluorescent particle is negligible during the counting interval T. In this case, the distribution of the number of photon counts emitted by a single fluorescent species can be analytically expressed as double Poissonian: the distribution of the number of particles of given species within this volume is Poissonian, and the conditional probability of the number of detected photons corresponding to a given number of particles is also Poissonian. The double Poissonian distribution has two parameters: the mean number of particles in the measurement volume, c and the mean number of photons emitted by a single particle per dwell time, qT. The distribution of the number of photon counts n corresponding to a single species is expressed as                                           P            ⁡                          (                                                n                  ;                  c                                ,                q                            )                                =                                    ∑                              n                =                0                            ∞                        ⁢                                                            c                  m                                                  m                  !                                            ⁢                              ⅇ                                  -                  c                                            ⁢                                                                    (                    mqT                    )                                    n                                                  n                  !                                            ⁢                              ⅇ                                  -                  mqT                                                                    ,                            (        6        )            
where m runs over the number of molecules in the measurement volume. If Pi(n) denotes the distribution of the number of photon counts from species i, then the resultant distribution P(n) is expressed as                               P          ⁡                      (            n            )                          =                              ∑                          (                              n                1                            )                                ⁢                                    ∏              l                        ⁢                                                            P                  l                                ⁡                                  (                                      n                    l                                    )                                            ⁢                              δ                ⁡                                  (                                      n                    ,                                                                  ∑                        i                                            ⁢                                              n                        i                                                                              )                                                                                        (        7        )            
This means that P(n) can be calculated as a convolution of the series of distribution Pi(n).
Like in FCS, the rectangular sample profile is a theoretical model which can hardly be applied in experiments. One may divide the measurement volume into a great number of volume elements and assume that within each of them, the intensity of a molecule is constant. Contribution to photon count number distribution from a volume element is therefore double Poissonian with parameters cdV and qTB(r). (Here q denotes count rate from a molecule in a selected standard position where B=1, and B(r) is the spatial brightness profile function of coordinates). The overall distribution of the number of photon counts can be expressed as a convolution integral over double Poissonian distributions. Integration is a one-dimensional rather than a three-dimensional problem here, because the result of integration does not depend on actual positions of volume elements in response to each other. Figuratively, one may rearrange the three-dimensional array of volume elements into a one-dimensional array, for example in the decreasing order of the value of B.
In a number of first experiments described in the international patent application PCT/EP 97/05619, the photon count number distribution was indeed fitted, using the convolution technique. The sample model consisted of twenty spatial sections, each characterized by its volume Vj and brightness Bj. However, the technique described in this patent application is slow and inconvenient in cases involving a high number of samples to be analyzed, like in diagnostics or drug discovery, or in analyzing distribution functions involving more than a single argument.
Therefore, it is an object of the present invention to present a convenient and much faster technique for analyzing fluorescence intensity fluctuations.
According to the present invention there is provided a method for characterizing fluorescent molecules or other particles in samples, the method comprising the steps of:
a) monitoring fluctuating intensity of fluorescence emitted by the molecules or other particles in at least one measurement volume of a non-uniform spatial brightness profile by measuring numbers of photon counts in primary time intervals by a single or more photon detectors,
b) determining at least one distribution function of numbers of photon counts, {circumflex over (P)}(n), from the measured numbers of photon counts,
c) determining physical quantities characteristic to said particles by fitting the experimentally determined distribution function of numbers of photon counts, wherein the fitting procedure involves calculation of a theoretical distribution function of the number of photon counts P(n) through its generating function, defined as       G    ⁡          (              ξ        _            )        =            ∑      n        ⁢                            ξ          _                n            ⁢                        P          ⁡                      (            n            )                          .            
The formal definition of the generating function of a distribution P(n) is as follows:                               G          ⁡                      (            ξ            )                          =                              ∑                          n              =              0                        ∞                    ⁢                                    ξ              n                        ⁢                                          P                ⁡                                  (                  n                  )                                            .                                                          (        8        )            
What makes the generating function attractive in count number distribution analysis is the additivity of its logarithm: logarithms of generating functions of photon count number distributions of independent sources, like different volume elements as well as different species, are simply added for the calculation of the generating function of the combined distribution because the transformation (8) maps distribution convolutions into the products of the corresponding generating functions.
In a particular preferred embodiment, one might monitor the fluctuating fluorescence intensity in consecutive primary time intervals of equal width. Typical primary time intervals have a width in order of several tenth of microseconds. The total data collection time is usually several tenth of seconds.
In a further preferred embodiment, numbers of photon counts {ni} subject to determination of a distribution function {circumflex over (P)}(n) in step b) are derived from numbers of photon counts in primary time intervals {Nj} by addition of numbers of photon counts from primary time intervals according to a predetermined rule. One might e.g. be interested in choosing numbers of photon counts {ni} subject to determination of a distribution function {circumflex over (P)}(n) which are calculated from the numbers of photon counts in primary time intervals {Nj} according to the rule             n      i        =                  ∑                  k          =          1                M            ⁢              N                  Mi          +          k                      ,
where M is an integer number expressing how many times the time interval in which {ni} is determined is longer than the primary time interval.
In a further embodiment, numbers of photon counts {ni} are derived from predetermined primary time intervals according to a rule in which primary time intervals are separated by a time delay. In particular, the following rule can be applied:             n      i        =                  ∑                  k          =          1                M            ⁢              (                              N                          Mi              +              k                                +                      N                                          M                ⁢                                  (                                      i                    +                    L                                    )                                            +              k                                      )              ,
where M and L are positive integer numbers, {ni} are numbers of photon counts subject to determination of a distribution function {circumflex over (P)}(n), and {Nj} are the numbers of photon counts in primary time intervals.
In some cases, it might be preferred to determine not only a single distribution function in step b), but rather a set of distribution functions {circumflex over (P)}(n). These can be determined according to a set of different rules, said set of distribution functions being fitted jointly in step c). As an example a set of distribution functions with different values of M and/or L might be fitted jointly.
Typical physical quantities which might be determined in step c) according to the present invention are concentration, specific brightness and/or diffusion coefficient of molecules or other particles.
In a further preferred embodiment, the generating function is calculated using the expression
G("xgr")=exp[∫dqc(q)∫d3r(e("xgr"xe2x88x921)qTB(r)xe2x88x921)],
where c(q) is the density of particles with specific brightness q, T is the length of the counting interval, and B(r) is the spatial brightness profile as a function of coordinates.
Applying the definition (8) to formula (6) with cxe2x86x92cdV and qxe2x86x92qB(r), the contribution from a particular species and a selected volume element dV can be written as
Gi("xgr";dV)=exp[cidV(cidV(e("xgr"xe2x88x921)qrTB(r)xe2x88x921)]. xe2x80x83xe2x80x83(9) 
Therefore, the generating function of the total photon count number distribution can be expressed in a closed form                               G          ⁡                      (            ξ            )                          =                              exp            ⁡                          [                                                ∑                  i                                ⁢                                                      c                    i                                    ⁢                                      ∫                                                                  (                                                                              ⅇ                                                                                          (                                                                  ξ                                  -                                  1                                                                )                                                            ⁢                                                              q                                1                                                            ⁢                                                              TB                                ⁡                                                                  (                                  r                                  )                                                                                                                                              -                          1                                                )                                            ⁢                                              ⅆ                        V                                                                                                        ]                                .                                    (        10        )            
Numeric integration according to Eq. (9) followed by a fast Fourier transform is the most effective means of calculating the theoretical distribution P(n) corresponding to a given sample (i.e., given concentrations and specific brightness values of fluorescent species). If one selects "xgr"=e1p, then the distribution P(n) and its generating function G(xcfx86) are interrelated by the Fourier transform. Therefore, it is particularly preferred to select the argument of the generating function in the form "xgr"=exe2x88x921p and to use a fast Fourier transform algorithm in calculation of the theoretical distribution of the number of photon counts out of its generating function.
When calculating the theoretical distribution P(n) in step c) according to the present invention, the spatial brightness profile might be modeled by a mathematical relationship between volume and spatial brightness. In particular, one might model the spatial brightness profile by the following expression:                     ⅆ        V                    ⅆ        x              =                            a          1                ⁢        x            +                        a          2                ⁢                  x          2                    +                        a          3                ⁢                  x          3                      ,
where dV denotes a volume element, x denotes logarithm of the relative spatial brightness, and xcex11, xcex12 and xcex13 are empirically estimated parameters.
Some fluorescent species may have a significantly wide distribution of specific brightness. For example vesicles, which are likely to have a significantly broad size distribution and a random number of receptors, may have trapped a random number of labeled ligand molecules. In order to fit count number distributions for samples containing such kind of species, it is useful to modify Eq. (10) in the following manner. The assumption is made that the distribution of brightness of particles q within a species is mathematically expressed as follows:
p(q)xe2x88x9dqaxe2x88x921exe2x88x92ba. xe2x80x83xe2x80x83(11) 
This expression has been selected for the sake of convenience: all moments of this distribution can be analytically calculated, using the following formula:                                           ∫            0            ∞                    ⁢                                    x              a                        ⁢                          ⅇ                              -                bx                                      ⁢                          ⅆ              x                                      =                                            Γ              ⁡                              (                                  a                  +                  1                                )                                                    b                              a                +                1                                              .                                    (        12        )            
It is straightforward to derive the modified generating function of a photon count number distribution. One can rewrite Eq. (9) as follows:                                           G            ⁡                          (              ξ              )                                =                      exp            ⁡                          [                                                ∑                  i                                ⁢                                                      c                    i                                    ⁢                                      ∫                                                                  ⅆ                        V                                            ⁢                                                                        ∫                          0                          ∞                                                ⁢                                                                              ⅆ                            q                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      ρ                            ⁡                                                          (                                                                                                q                                  ;                                                                      a                                    i                                                                                                  ,                                                                  b                                  i                                                                                            )                                                                                ⁢                                                      (                                                                                          ⅇ                                                                                                      (                                                                          ξ                                      -                                      1                                                                        )                                                                    ⁢                                                                      qTB                                    ⁡                                                                          (                                      r                                      )                                                                                                                                                                  -                              1                                                        )                                                                                                                                                          ]                                      ,                            (        13        )            
where                               ρ          ⁡                      (                                          q                ;                a                            ,              b                        )                          =                                            b              a                                      Γ              ⁡                              (                a                )                                              ⁢                      q                          n              -              1                                ⁢                                    ⅇ                              -                bq                                      .                                              (        14        )            
The integral over q can be performed analytically:                                           G            ⁡                          (              ξ              )                                =                      exp            ⁢                          {                                                ∑                  i                                ⁢                                                      c                    i                                    ⁢                                      ∫                                          ⅆ                                              V                        ⁡                                                  [                                                                                                                    (                                                                                                      b                                    i                                                                                                                                              b                                      i                                                                        -                                                                                                                  (                                                                                  ξ                                          -                                          1                                                                                )                                                                            ⁢                                                                              TB                                        ⁡                                                                                  (                                          x                                          )                                                                                                                                                                                                                    )                                                                                            a                                1                                                                                      -                            1                                                    ]                                                                                                                                }                                      ,                            (        15        )            
The parameters xcex1i and bi are related to the mean brightness {overscore (q)}i and the width of the brightness distribution "sgr"i2 by                                           a            i                    =                                                    q                _                            i              2                                      σ              i              2                                      ,                  xe2x80x83                ⁢                              b            i                    =                                                                      q                  _                                i                                            σ                i                2                                      .                                              (        16        )            
In the range of obtained count numbers, the probability to obtain a particular count number usually varies by many orders of magnitude, see for example the distribution of FIG. 1. Consequently, the variance of the number of events with a given count number has a strong dependence on the count number. To determine weights for least squares fitting, one may assume that light intensities in all counting intervals are independent. Under this assumption, one has a problem of distributing M events over choices of different count numbers n, each particular outcome having a given probability of realization, P(n). Covariance matrix elements of the distribution can be expressed as follows:                                           ⟨                          Δ              ⁢                              xe2x80x83                            ⁢                              P                ⁡                                  (                  n                  )                                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢                              P                ⁡                                  (                  m                  )                                                      ⟩                    =                                                                      P                  ⁡                                      (                    n                    )                                                  ⁢                                  δ                  ⁡                                      (                                          n                      ,                      m                                        )                                                              -                                                P                  ⁡                                      (                    n                    )                                                  ⁢                                  P                  ⁡                                      (                    m                    )                                                                        M                          ,                            (        17        )            
where M is the number of counting intervals per experiment.
For a further simplification, one may ignore the second term on the right side of Eq. (17), which can be interpreted as a consequence of normalization. In this case, the weights simply equal to the inverse values of the diagonal covariance matrix elements                               W          n                =                              M                          P              ⁡                              (                n                )                                              .                                    (        18        )            
Dispersion matrix (17) corresponds to the multinomial distribution of statistical realizations of histograms. The Poissonian distribution, with the constraint that the total number of counting intervals M is fixed, will lead to the multinomial distribution. This is the rationale behind using Poissonian weights as given in Eq. (18).
Let nk be the expectation value of the number of events of counting k photons and let   N  =            ∑      k        ⁢          n      k      
be their sum. Let mk be a statistical realization with   M  =            ∑      k        ⁢                  m        k            .      
Assume that realizations m0,m1, . . . obey Poissonian statistics                                           P            (                                          m                0                            ,                              m                1                            ,              …                        ⁢                          xe2x80x83                        )                    =                                                                      [                                                            n                      0                                        ,                                          n                      1                                        ,                    …                                    ⁢                                      xe2x80x83                                    ]                                                  [                                                            m                      0                                        ,                                          m                      1                                        ,                    …                                    ⁢                                      xe2x80x83                                    ]                                                                              [                                                            m                      0                                        ,                                          m                      1                                        ,                    …                                    ⁢                                      xe2x80x83                                    ]                                !                                      ⁢                          ⅇ                              -                N                                                    ,                            (        19        )            
where we have introduced the notation
n0m0n1m1 . . . xe2x89xa1[n0,n1, . . . ][m0,m1, . . . ]
and
m0!m1! . . . xe2x89xa1[m0,m1, . . . ].
The probability of having the total of M events is                               P          ⁡                      (            M            )                          =                                            N              M                                      N              !                                ⁢                                    ⅇ                              -                N                                      .                                              (        20        )            
The conditional probability of having m0,m1, . . . events if there is a total of M events is                     P        ⁡                  (                                    m              0                        ,                          m              1                        ,                          …              |              M                                )                    ≡                        P          (                                    m              0                        ,                          m              1                        ,            …                    ⁢                      xe2x80x83                    )                          P          ⁡                      (            M            )                                =                                        M            !                    [                                    n              0                        ,                          n              1                        ,            …                    ⁢                      xe2x80x83                    ]                          [                                    m              0                        ,                          m              1                        ,            …                    ⁢                      xe2x80x83                    ]                                                  N            M                    [                                    m              0                        ,                          m              1                        ,            …                    ⁢                      xe2x80x83                    ]                !              ,
or
P(m0,m1, . . . 1M)xe2x89xa1Cm0,m1, . . . M[p0,p1, . . . ][m0,m1, . . . ], xe2x80x83xe2x80x83(21) 
This is the multinomial distribution where we have introduced:       p    k    ≡            n      k        M  
and Cm0,m1, . . . M are multinomial coefficients.
In general, a linear or linearized least squares fitting returns not only the values of the estimated parameters, but also their covariance matrix, provided the weights have been meaningfully set. It may turn out to be possible to express the statistical errors of the estimated parameters analytically in some simple cases (e.g., for the rectangular sample profile and single species) but in applications at least two-component analysis is usually of interest. Therefore, one may be satisfied with the numerical calculations of statistical errors. In addition to the xe2x80x9ctheoreticalxe2x80x9d errors with the assumption of non-correlated measurements (Eq. (17)), in some cases statistical errors have been estimated empirically, making a series of about a hundred FIDA experiments on identical conditions. As a rule, empirical errors are higher than theoretical ones by a factor of three to four. Empirical errors appear to be closer to the theoretical ones in scanning experiments. Therefore we are convinced that the main reason of the underestimation of theoretical errors is the assumption of non-correlated measurements. Table 1 compares statistical errors of parameters estimated by fitting a photon count number distribution (FIDA) according to the present invention and by the moment analysis. Error values are determined through processing a series of simulated distributions. The present invention is overwhelmingly better than the moment analysis if the number of estimated parameters is higher than three.
Confocal techniques may be applied to a wide field of applications, such as biomedicine, diagnostics, high through-put drug screening, sorting processes such as sorting of particles like beads, vesicles, cells, bacteria, viruses, etc. The conjugate focal (confocal) technique is based on using a point source of light sharply focused to a diffraction-limited spot on the sample. The emitted light is viewed through a spatial filter (pinhole) that isolates the viewing area to that exactly coincident with the illuminating spot. Thus, the illumination and detection apertures are optically conjugated with each other. Light originating from focal planes other than that of the objective lens is rejected, which effectively provides a very small depth of field. Therefore, in a particular preferred embodiment of the present invention, in step a) a confocal microscope is used for monitoring the intensity of fluorescence. In order to achieve a high signal-to-noise ratio, it is useful to monitor the intensity of fluorescence using an apparatus that comprises: a radiation source (12) for providing excitation radiation (14), an objective (22) for focussing the excitation radiation (14) into a measurement volume (26), a detector (42) for detecting emission radiation (30) that stems from the measurement volume (26), and an opaque means (44) positioned in he pathway (32) of the emission radiation (30) or excitation radiation (14) for erasing the central part of the emission radiation (30) or excitation radiation (14). It might be particularly preferred to use an optical set-up described in detail in FIG. 9.
In a further preferred embodiment, the method according to the present invention is applied to fit a joint distribution of photon count numbers. In experiments, fluorescence from a microscopic volume with a fluctuating number of molecules is monitored using an optical set-up (e.g. a confocal microscope) with two detectors. The two detectors may have different polarizational or spectral response. In one embodiment, concentrations of fluorescent species together with two specific brightness values per each species are determined. The two-dimensional fluorescence intensity distribution analysis (2D-FIDA) if used with a polarization cube is a tool which can distinguish fluorescent species with different specific polarization ratios. This is a typical example of a joint analysis of two physical characteristics of single molecules or other particles, granting a significantly improved reliability compared to methods focussed on a single physical characteristic.
In order to express the expected two-dimensional distribution of the number of photon counts, it is favourable to use the following assumptions: (A) Coordinates of particles are random and independent of each other. (B) Contribution to fluorescence intensity from a particle can be expressed as a product of a specific brightness of the particle and a spatial brightness profile function characteristic to the optical equipment. (C) A short counting time interval T is selected, during which brightness of fluorescent particles does not significantly change due to diffusion.
At first, a joint distribution of count numbers from a single fluorescent species and a single small open volume element dV is expressed. The volume element is characterized by coordinates r and spatial brightness B(r), and the fluorescent species is characterized by its specific brightness values q1 and q2. By q1 and q2, mean photon count rates by two detectors from a particle situated at a point where B(r)=1 are denoted. A convenient choice is to select a unit of B, as usual in FCS, by the equation X1=X2, where X1=∫Bk(r)d3r. If the volume element happens to contain m particles, then the expected photon count numbers per time interval T from the volume element are mq1TB(r) and mq2TB(r), while the distribution of numbers of photon counts from m particles P(n1,n2|m) is Poissonian for both detectors independently:                               P          ⁡                      (                                          n                1                            ,                                                n                  2                                |                m                                      )                          =                                                            (                                                      mq                    1                                    ⁢                                      TB                    ⁡                                          (                      x                      )                                                                      )                                            n                1                                                                    n                1                            !                                ⁢                      ⅇ                                          -                                  mq                  1                                            ⁢                              TB                ⁡                                  (                  x                  )                                                              ⁢                                                    (                                                      mq                    2                                    ⁢                                      TB                    ⁡                                          (                      x                      )                                                                      )                                            n                2                                                                    n                2                            !                                ⁢                                    ⅇ                                                -                                      mq                    2                                                  ⁢                                  TB                  ⁡                                      (                    x                    )                                                                        .                                              (        22        )            
From the other side, under assumption (A), the distribution of the number of particles of given species in the volume element is Poissonian with mean cdV, c denoting concentration:                                           P            dV                    ⁡                      (            m            )                          =                                                            (                cdV                )                            m                                      m              !                                ⁢                                    ⅇ                              -                edV                                      .                                              (        23        )            
The overall distribution of the number of photon counts from the volume element can be expressed using Eqs.22 and 23:                                                                                           P                  dV                                ⁡                                  (                                                            n                      1                                        ,                                          n                      2                                                        )                                            =                              xe2x80x83                            ⁢                                                ∑                  m                                ⁢                                                                            P                      dV                                        ⁡                                          (                      m                      )                                                        ⁢                                      P                    ⁡                                          (                                                                        n                          1                                                ,                                                                              n                            2                                                    |                          m                                                                    )                                                                                                                                              =                              xe2x80x83                            ⁢                                                ∑                  m                                ⁢                                                                                                    (                        cdV                        )                                            m                                                              m                      !                                                        ⁢                                      ⅇ                                          -                      edV                                                        ⁢                                                                                    (                                                                              mq                            1                                                    ⁢                                                      TB                            ⁡                                                          (                              r                              )                                                                                                      )                                                                    n                        1                                                                                                            n                        1                                            !                                                        ⁢                                      ⅇ                                                                  -                                                  mq                          1                                                                    ⁢                                              TB                        ⁡                                                  (                          r                          )                                                                                                                                                                                                            xe2x80x83                            ⁢                                                                                          (                                                                        mq                          2                                                ⁢                                                  TB                          ⁡                                                      (                            r                            )                                                                                              )                                                              n                      2                                                                                                  n                      2                                        !                                                  ⁢                                  ⅇ                                                            -                                              mq                        2                                                              ⁢                                          TB                      ⁡                                              (                        r                        )                                                                                                                                                    (        24        )            
As in the one-dimensional case described above, a useful representation of a distribution of numbers of photon counts P(n1,n2) in its generating function, defined as                               G          ⁡                      (                                          ξ                1                            ,                              ξ                2                                      )                          =                              ∑                                          n                1                            =              0                        m                    ⁢                                    ∑                                                n                  2                                =                0                            ∞                        ⁢                                          ξ                1                                  n                  1                                            ⁢                              ξ                2                                  n                  2                                            ⁢                                                P                  ⁡                                      (                                                                  n                        1                                            ,                                              n                        2                                                              )                                                  .                                                                        (        25        )            
The generating function of the distribution expressed by Eq 24 can be written as                                                                                           G                  dV                                ⁡                                  (                                                            ξ                      1                                        ,                                          ξ                      2                                                        )                                            =                              xe2x80x83                            ⁢                                                ⅇ                                      -                    cdV                                                  ⁢                                                      ∑                    m                                    ⁢                                                                                                              (                          cdV                          )                                                m                                                                    m                        !                                                              ⁢                                          ⅇ                                                                        -                                                      mq                            1                                                                          ⁢                        BT                                                              ⁢                                          ⅇ                                                                        -                                                      mq                            2                                                                          ⁢                        BT                                                              ⁢                                                                  ∑                                                  n                          1                                                                    ⁢                                                                                                    (                                                          m                              ⁢                                                              xe2x80x83                                                            ⁢                                                              ξ                                1                                                            ⁢                              qBT                                                        )                                                                                n                            1                                                                                                                                n                            1                                                    !                                                                                                                                                                                                            xe2x80x83                            ⁢                                                ∑                                      n                    2                                                  ⁢                                                                            (                                              m                        ⁢                                                  xe2x80x83                                                ⁢                                                  ξ                          2                                                ⁢                        qBT                                            )                                                              n                      2                                                                                                  n                      2                                        !                                                                                                                          =                              xe2x80x83                            ⁢                                                ⅇ                                      -                    cdV                                                  ⁢                                                      ∑                    m                                    ⁢                                                                                    {                                                  cdV                          ⁢                                                      xe2x80x83                                                    ⁢                                                      exp                            ⁡                                                          [                                                                                                (                                                                                                            ξ                                      1                                                                        -                                    1                                                                    )                                                                ⁢                                                                  q                                  1                                                                ⁢                                BT                                                            ]                                                                                ⁢                                                      exp                            ⁡                                                          [                                                                                                (                                                                                                            ξ                                      2                                                                        -                                    1                                                                    )                                                                ⁢                                                                  q                                  2                                                                ⁢                                BT                                                            ]                                                                                                      }                                            m                                                              m                      !                                                                                                                                              =                              xe2x80x83                            ⁢                              exp                [                                  cdV                  ⁡                                      (                                                                                            ⅇ                                                                                    (                                                                                                ξ                                  1                                                                -                                1                                                            )                                                        ⁢                                                          q                              1                                                        ⁢                            BT                                                                          ⁢                                                  ⅇ                                                                                    (                                                                                                ξ                                  2                                                                -                                1                                                            )                                                        ⁢                                                          q                              2                                                        ⁢                            BT                                                                                              -                      1                                        )                                                  }                                                                        (        26        )            
In particular, if one selects "xgr"k=exp(ixcfx86k), then the distribution P(n1,n2) and its generating function G(xcfx861,xcfx862) are interrelated by a 2-dimensional Fourier transform. What makes the generating function attractive in photon count number distribution analysis is the additivity of its logarithm: logarithms of generating functions of photon count number distributions of independent sources, like different volume elements as well as different species, are simply added for the calculation of the combined distribution. Therefore, the generating function of the overall distribution of the number of photon counts can be expressed in a closed form:                               G          ⁡                      (                                          ξ                1                            ,                              ξ                2                                      )                          =                              exp            [                                                            (                                                            ξ                      1                                        -                    1                                    )                                ⁢                                  λ                  1                                ⁢                T                            +                                                (                                                            ξ                      2                                        -                    1                                    )                                ⁢                                  λ                  2                                ⁢                T                            +                                                ∑                  i                                ⁢                                                      c                    i                                    ⁢                                      ∫                                                                  (                                                                              ⅇ                                                                                                                            (                                                                                                            ξ                                      1                                                                        -                                    1                                                                    )                                                                ⁢                                                                  q                                  1                                                                                            ,                                                                                                TB                                  ⁡                                                                      (                                    r                                    )                                                                                                  ⁢                                                                  (                                                                                                            ξ                                      2                                                                        -                                    1                                                                    )                                                                ⁢                                                                  q                                  2                                                                                            ,                                                              TB                                ⁡                                                                  (                                  r                                  )                                                                                                                                              -                          1                                                )                                            ⁢                                                                        ⅆ                          3                                                ⁢                        r                                                                                                                  ]                    .                                    (        27        )            
In this formula, a contribution from background count rates, xcex1 by detector 1 and xcex2 by detector 2, as well as contributions from different fluorescent species, denoted by the subscript i, have been integrated. Numeric integration according to Eq 27 followed by a fast Fourier transform is a very efficient means for calculation of the theoretical distribution P(n1,n2) corresponding to a given sample (i.e. given concentrations and specific brightness values of fluorescent species).
The spatial brightness function is accounted through the spatial integration on the right side of Eq. 27. The three-dimensional integration can be reduced to a one-dimensional one by replacing three-dimensional coordinates r by a one-dimensional variable, a monotonic function of the spatial brightness B(r). A convenient choice of the variable is x=1n[B(0)/B(r)]. A sufficiently flexible model of the spatial brightness profile is presented by the following expression:                                           ⅆ            V                                ⅆ            x                          ∝                              x            ⁡                          (                              1                +                                                      a                    1                                    ⁢                  x                                +                                                      a                    2                                    ⁢                                      x                    2                                                              )                                .                                              (          28          )                ⁢                  xe2x80x83                    
In the interval of obtained count numbers, the probability to obtain a particular pair of count numbers usually varies by many orders of magnitude. Consequently, the variance of the experimental distribution has also a strong dependence on the count numbers. To determine weights for least squares fitting, for simplification it is assumed that coordinates of particles in all counting intervals are randomly selected. (This means one ignores correlations of the coordinates in consecutive counting intervals.) Under this assumption, one has a problem of distributing M events over choices of different pairs of count numbers n1, n2, each particular outcome having a given probability of realization, P(n1,n2). Covariance matrix elements of the distribution can be expressed as follows:                                           ⟨                          Δ              ⁢                              xe2x80x83                            ⁢                              P                ⁡                                  (                                                            n                      1                                        ,                                          n                      2                                                        )                                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢                              P                ⁡                                  (                                                            n                      1                      xe2x80x2                                        ,                                          n                      2                      xe2x80x2                                                        )                                                      ⟩                    =                                                                      P                  ⁡                                      (                                                                  n                        1                                            ,                                              n                        2                                                              )                                                  ⁢                                  δ                  ⁡                                      (                                                                  n                        1                                            ,                                              n                        1                        xe2x80x2                                                              )                                                  ⁢                                  δ                  ⁡                                      (                                                                  n                        2                                            ,                                              n                        2                        xe2x80x2                                                              )                                                              -                                                P                  ⁡                                      (                                                                  n                        1                                            ,                                              n                        2                                                              )                                                  ⁢                                  P                  ⁡                                      (                                                                  n                        1                        xe2x80x2                                            ,                                              n                        2                        xe2x80x2                                                              )                                                                        M                          ,                            (        29        )            
where M is the number of counting intervals per experiment.
For a further simplification, one may ignore the second term on the right side of Eq. (29), which can be interpreted as a consequence of normalization. In this case, the weights simply equal to the inverse values of the diagonal covariance matrix elements                               W          ⁡                      (                                          n                1                            ,                              n                2                                      )                          =                              M                          P              ⁡                              (                                                      n                    1                                    ,                                      n                    2                                                  )                                              .                                    (        30        )            
In general, a linearized least squares fitting algorithm returns not only values of estimated parameters, but also their covariance matrix, provided weights have been meaningfully set. In addition to xe2x80x9ctheoreticalxe2x80x9d errors corresponding to the assumption of uncorrelated measurements (Eq. (30)), in some cases statistical errors have been determined empirically, making a series of about 100 2D-FIDA experiments at identical conditions. As a rule, empirical errors are higher than theoretical ones by a factor of three to four. The main reason of underestimation of theoretical errors is most likely the assumption of uncorrelated measurements.
Even though the assumption of uncorrelated measurements yields underestimated error values, it is a very useful theoretical approximation, allowing to compare accuracy of analysis under different experimental conditions as well as different methods of analysis. Also, this approximation provides an easy method of data simulation which is a useful tool in general. A simple and very fast method of data simulation is calculation of the expected event number as a function of photon count numbers and addition of random Poisson noise to each event number independently.
In Table 2 theoretical errors of one-dimensional and two-dimensional fluorescence intensity distribution analysis according to the present invention are presented in two selected cases of two fluorescent species. In both cases the ratio of specific brightness values of the two species is three. In the case of 2D-FIDA, it is assumed that spectral sensitivities of the two detectors are tuned to different species. In both cases data collection time of 10 s, time window of 40 xcexcs and background count rate of 1 kHz are assumed. Note that the statistical errors of the estimated parameters are significantly lower in the 2D-FIDA example.
The two-dimensional fluorescent intensity distribution analysis according to the present invention is in the following compared to the prior art moment analysis.
Factorial moments of the distribution P(n1,n2) are defined as                               F          kl                =                              ∑                                          n                1                            ,                              n                2                                              ⁢                                                                                          n                    1                                    !                                ⁢                                                      n                    2                                    !                                                                                                  (                                                                  n                        1                                            -                      k                                        )                                    !                                ⁢                                                      (                                                                  n                        2                                            -                      l                                        )                                    !                                                      ⁢                                          P                ⁡                                  (                                                            n                      1                                        ,                                          n                      2                                                        )                                            .                                                          (        31        )            
Factorial moments are related to factorial cumulants Kk1                               K          kl                =                              F            kl                    -                                    ∑                                                i                  ,                  j                                                                      i                    +                    j                                     greater than                   0                                                      ⁢                                          C                i                                  k                  +                  1                                            ⁢                              C                j                l                            ⁢                              K                                                      k                    -                    i                                    ,                                      l                    -                    j                                                              ⁢                                                F                  lj                                .                                                                        (        32        )            
C denotes binomial coefficients. Cumulants can be expressed through concentrations and specific brightness values by a simple relation                               K          kl                =                              χ                          k              +              1                                ⁢                      T                          k              +              1                                ⁢                                    ∑              l                        ⁢                                          c                i                            ⁢                              q                                  1                  ⁢                  i                                k                            ⁢                                                q                                      2                    ⁢                    i                                    l                                .                                                                        (        33        )            
If the unit of B is selected by the equation X1=X2, Z1 has the meaning of the sample volume, denoted by V, and Eq. 33 can be written as                                           γ                          k              +              1                                ⁢                      K            kl                          =                              ∑            l                    ⁢                                    (                                                c                  i                                ⁢                V                            )                        ⁢                                          (                                                      q                                          1                      ⁢                      i                                                        ⁢                  T                                )                            k                        ⁢                                                            (                                                            q                                              2                        ⁢                        i                                                              ⁢                    T                                    )                                l                            .                                                          (        34        )            
where xcex3 denotes a series of constants characterizing the brightness profile:                               γ          m                =                                            χ              l                                      χ              m                                .                                    (        35        )            
The principle of moment analysis is to determine values of a few cumulants from an experiment and solve a system of Eqs. 33 in respect to unknown concentrations and brightness values.
In Table 3 statistical errors of 2D-FIDA (present invention) and 2D-MAFID (prior art) are presented determined by generating a series of 30 random distributions of count numbers, simulated for identical xe2x80x9csamplesxe2x80x9d, thereafter applying 2D-FIDA and 2D-MAFID, and determining the variance of estimated parameters in both cases. Note a tendency that the advantages of 2D-FIDA compared to 2D-MAFID increase with the number of parameters to be estimated.
Error values are calculated from the scattered results of analysis applied to a series of simulated data.
In Table 4 the relative deviation of mean values (i.e. bias) of estimated parameters are presented for 2D-FIDA (present invention) and 2D-MAFID (prior art). In each case bias is determined from analysis of a series of thirty simulated random distributions of count numbers. Three cases were analysed. In the first case, models used in data simulations and data analysis were identical. In the second case, the distributions of count numbers were simulated assuming that particles of the second species are not equivalent but being distributed by their individual brightness with a relative half-width of 20 percent. This phenomenon was intentionally ignored in analysis, however. Of course, applying a slightly inadequate model for analysis produces bias of estimated parameters. The third case is similar to the second one except the relative half-width of the individual brightness distribution of the second species is 50 percent, which is a usual value for vesicular preparations. It is worth noting that methodological deviations are noticeable when mapping weighted residuals of 2D-FIDA in cases two and three, but 2D-FIDA still returns meaningful results, 2D-MAFID is a more sensitive method to model roughness.